Contribution 1: We provide an alternative characterization of
\(\mathcal {F}_{\infty -\text{mars}}^d\)  and \(V_{\mathrm {mars}}(f)\)  in terms of the Hardy-Krause
variations of mixed partial derivatives of \(f\)  of orders up to 1.
Hardy-Krause variation [1]}, [2]} is a multivariate generalization of
total variation of univariate functions (we review basic
properties of Hardy-Krause variation in Section
REF ). The precise form of the connection between
\(V_{\mathrm {mars}}(\cdot )\)  and Hardy-Krause variation is given in Proposition
REF . Here let us just additionally note that when \(f\) 
is sufficiently smooth (in the sense of Lemma
REF ), we have
\(V_{\mathrm {mars}}(f) = \sum _{\begin{array}{c}\beta \in \lbrace 0, 1, 2\rbrace ^d \\ \max _j \beta _j = 2\end{array}}\int _{\bar{T}^{(\beta )}} |f^{(\beta )} |\) 
